Abstract Latouche and Nguyen (2015b) constructed a sequence of stochastic fluid processes showed that it converges weakly to Markov-modulated Brownian motion (MMBM). Here, we construct different show strongly an MMBM. To the best our knowledge, this is first result on strong convergence motion. Besides implying weak convergence, such approximation constitutes powerful tool for developing deep r...

Various consistency proofs for the kernel density estimator have been developed over the last few decades. Important milestones are the pointwise consistency and almost sure uniform convergence with a fixed bandwidth on the one hand and the rate of convergence with a fixed or even a variable bandwidth on the other hand. While considering global properties of the empirical distribution functions...